1. Field of the Invention
This invention pertains to bench blasting methods and, in particular, to a method of selecting the placement of boreholes along a drill line.
In modern bench blasting, vertical or near vertical holes are drilled adjacent to a rock face and are loaded with explosive charges that are then detonated. The detonation fractures the rock mass between the borehole and the rock face and displaces the resulting fractured rock. The resulting broken rock, known as xe2x80x9cmuckxe2x80x9d, is removed and a new free rock face is thus exposed. If the muck contains a desired product, it can be gathered and processed. Otherwise, it may simply be removed from the blasting site to permit further blasting or other activities.
2. Related Art
U.S. Pat. No. 3,377,909 to Grant et al, issued Apr. 16, 1968, discloses the use of a xe2x80x9cpowder factorxe2x80x9d (cubic yards of earth per pound of explosive) to characterize a borehole pattern in a coal field strip mine and discloses a xe2x80x9cnormalxe2x80x9d spacing for ANFO (see col. 7, lines 70-71 and 6, lines 63-68).
U.S. Pat. No. 3,848,927 to Livingston, dated Nov. 19, 1974, discloses a trial and error method of determining the optimum and critical depths of a small charge, and teaches the scaling-up of this information for larger charges based on cube root scaling (see col. 7, lines 6-45). This patent suggests matching the charge to the desired size of debris.
U.S. Pat. No. 4,273,049 to Edwards et al, dated Jun. 16, 1981, suggests over-coming the dampening effect of water in a borehole by using water-resistant explosive in the water-containing portions of the borehole and conventional explosives above those portions.
U.S. Pat. No. 4,440,447 to Ricketts et al, dated Apr. 3, 1984, teaches that, in a borehole array for the formation of a retort in oil shale, outer boreholes can be closely spaced and made smaller in diameter to maintain the powder factor (see col. 8, lines 40-53), which is defined as the ratio of energy or explosive used per unit volume of formation explosively expanded in pounds ANFO equivalent per ton of oil shale formation expanded (see column 10, lines 13-17). No explanation of the term xe2x80x9cANFO equivalentxe2x80x9d is given.
International Patent Application PCT/GB90/00567, which is incorporated herein by reference or background information, discloses a laser rangefinder device referred to by the trademark QUARRYMAN that can be used to survey a rock face and, when given a borehole pattern by the user, to calculate the burden associated with each borehole. This patent application also discloses a borehole analyzer referred to by the trademark BORETRAK that allows the user to determine the configuration of a borehole as actually drilled.
Prior art methods for assessing the rock hole burden associated with a given borehole or, alternatively, for predicting the optimum positions for boreholes along a rock face, made use only of gross approximations of the burdens associated with the boreholes. Typically, the volume of explosive material in the borehole is calculated and a known conversion factor corresponding to the powder factor disclosed in U.S. Pat. No. 3,377,909 (discussed above) is used to project a volume of rock to be associated with the explosive material in the borehole, i.e., the rock burden. The rock burden is then expressed as a roughly rectangular block, one dimension of which corresponds to the length of the column of explosive material in the borehole, another to the distance of the borehole to the rock face. The projected hole spacing along the drill line can then be derived as the third dimension of the rectangular block. This calculation method is highly inefficient because it does not take into account significant variations in the configuration of the rock face that can occur within the dimensions of the rectangular block associated with the borehole.
One broad aspect of the present invention pertains to a method for establishing a drill pattern for a plurality of boreholes of predetermined diameter for use with a specified explosive material along a drill line along a bench of rock having a known density and a rock face. The method comprises (a) defining a drill line having a start point and an end point; (b) determining a target rock burden BT for a hypothetical borehole having the predetermined diameter at the start point; (c) defining along the drill line a progression of successive layers of rock, each layer defining an incremental burden, and determining the cumulative burden Bcum of the successive layers and revising BT with each successive layer until Bcum accounts for one-half of BT; (d) setting and indicating a position for the borehole on the drill line in the most distant layer from the start point; (e) defining additional successive layers of rock until the total of the incremental burdens of the layers defined in steps (c) and (e) accounts for BT; (f) setting and indicating a location for a distant boundary of the rock burden for the borehole; and (g) using the distant boundary as the start point for an additional borehole and repeating steps (b), (c) (d), (e) and (f) for each additional borehole until a layer coincides with the end point.
In one example, such a method may comprise the foregoing steps (a) and (b) and then (c) defining along the drill line a progression of successive intermediate layers of rock each having a mass less than the target rock burden BT and being bounded by an intermediate boundary plane and a distant boundary plane and for each intermediate layer (i) calculating a revised BT based on a hypothetical borehole on the last defined boundary plane, and (ii) determining the cumulative burden Bcum of the defined incremental intermediate layers until Bcum accounts for one-half of BT and then setting and indicating the location of a borehole on the drill line in the last defined layer (referred to as the xe2x80x9ccentral layerxe2x80x9d); (d) defining along the drill line a progression of successive distant layers of rock, and accumulating the rock burdens of the layers until the total rock burden accumulated in steps (c) and (d) accounts for BT; (e) setting and indicating a location for a distant boundary of the rock burden for the borehole; and (f) using the distant boundary as a start point and returning to steps (b)-(e) until an incremental layer coincides with the end point.
Optionally, the position of the borehole may be set between the planar boundaries of the central layer by interpolation or on one of the boundaries.
Another aspect of this invention relates to a method for proposing a drill pattern comprising positions for boreholes of predetermined diameter for use with a specified explosive material along a drill line along a bench of rock having a known density and rock face. The method comprises (a) defining a drill line having a start point and an end point; (b) determining a target rock burden (BT) for a hypothetical borehole having a height corresponding to the start point; (c) defining along the drill line a progression of successive layers of rock each having a mass less than BT and each being bounded by planar cross sections of the bench and having an intermediate boundary plane and a distant boundary plane, determining the cumulative burden Bcum of the defined layers, and calculating an average height of the layers with each successive layer; (d) using the average height to calculate a revised BT for the hypothetical borehole; and (e) repeating steps (c) and (d) until Bcum accounts for BT and then indicating the location of a borehole on the drill line between the start point and the most distant layer, and using the distant boundary of the most distant layer as a start point and returning to step (b) until a layer coincides with the end point.
According to one aspect of the invention, calculating the average height of the incremental layers may comprise defining spaced parallel planes that define layer boundaries and taking the average height of the planes.
According to another aspect of the invention, the rock mass of a layer may be calculated as the rock density multiplied by the volume of the layer and the volume may be calculated as one-half of the sum of the surface areas of the planes bounding the layer multiplied by the spacing between the planes.
According to still another aspect of the invention, determining BT may comprise determining the amount of the specified explosive material that would be loaded in the hypothetical borehole, converting the amount to a corresponding quantity of a reference explosive material and calculating a target burden associated with the corresponding quantity of the reference explosive material. Optionally, converting the amount to a corresponding quantity of a reference explosive material may comprise scaling the mass of the specified explosive material by the relative magnitudes of the specific energies of the specified explosive material and the reference explosive material. In a particular embodiment, calculating the target burden may comprise determining a Material Factor for the reference explosive material and multiplying the corresponding quantity by the Material Factor.
According to yet another aspect of the invention, determining BT may comprise determining an Energy Factor for the rock burden and relating the rock burden to the amount of explosive material that would be in the hypothetical borehole using the Energy Factor.
The method of this invention may optionally include designating blast design constraints comprising minimum and maximum values for hole-to-rock face burden, hole spacing and at least one of a Material Factor and Energy Factor and determining and indicating for each borehole whether the constraints are met. Optionally, the drill pattern characteristics of each borehole may be determined on a section-by-section basis for each borehole. The method may further include analyzing deviations of drill pattern characteristics from the constraints to evaluate at least one of the drill line distance and drill line orientation relative to the rock face and indicating the evaluation. Optionally, the method may include identifying and reporting each borehole having an excess toe burden or a swell or a hollow in the rock face.
The present invention further provides a method for determining a priority-directed loading configuration for a borehole subject to blast design criteria. This method comprises (a) selecting a segment of the explosive column portion of the borehole to be filled with explosive material; (b) determining the rock burden associated with the identified segment; (c) evaluating candidate explosive materials for use in the identified segment in order of priority until one is found that meets the blast design criteria (referred to herein as a compliant material), and assigning the first compliant material to the selected segment, assigning stemming to the segment when all the candidate explosive materials fail to meet the minimum energy factor criterion and indicating xe2x80x9cunknownxe2x80x9d when all the candidate explosive materials exceed the maximum energy factor criterion; and (d) repeating steps (a), (b) and (c) for each segment of the explosive column. Optionally, the explosive materials may be evaluated in order of cost to generate a cost-directed loading configuration. Alternatively, the explosive materials may be evaluated in order of specific energy. In a particular embodiment, the method may comprise assigning stemming to the segment when each evaluated explosive material provides less than a minimum energy factor criterion for the rock burden. The method may optionally further comprise indicating whether all candidate explosive materials exceed the maximum energy criterion for a segment of the borehole.
The present invention also provides a method for choosing at least one explosive material for use in at least a segment of a borehole, the method comprising determining a target specific volume energy required for an explosive material in the borehole; and identifying at least one explosive material that provides at least the target specific volume energy. This method of the present invention may comprise comparing the specific energies of candidate explosive materials to the target specific volume energy, which may optionally comprise referring to stored data that indicate specific volume energies of a plurality of explosive materials. The stored data may indicate the densities and specific mass energies of the various candidate explosive materials, and identifying an explosive material may comprise calculating the specific volume energy of a candidate explosive material and comparing the candidate specific volume energy to the target specific volume energy.
Optionally, the method may comprise partitioning the borehole into segments and determining rock burden and target specific volume energy for various segments of the borehole and separately identifying an explosive material for each segment. In a particular embodiment, the method may further comprise determining the rock burden for the borehole and using a predetermined Energy Factor and the size of the borehole to determine the corresponding specific volume energy.
Another method of the present invention relates to evaluating the suitability of a candidate explosive material of known specific energy for use in at least a segment of a borehole having a predetermined diameter and having a rock burden associated therewith by identifying a reference Material Factor (MFR) for the rock burden associated with the borehole with reference to a reference explosive material of known specific energy; calculating an adjusted Material Factor (MFA) corresponding to the use of the candidate explosive material in the borehole; and comparing the adjusted Material Factor (MFA) to the reference Material Factor (MFR). Calculating the adjusted Material Factor (MFA) may comprise multiplying the reference Material Factor (MFR) by (MRef)(ERef)/(MEXP)(EEXP); wherein Mref=the mass of reference explosive in the section of the borehole; MEXP=the mass of candidate explosive in the section of the borehole; ERef=the specific mass energy of the reference explosive; and EEXP=the specific mass energy of the candidate explosive, so that MFA=MFR ((MRef)(ERef)/(MEXP)(EEXP)).
The present invention also provides a method for selecting the diameter of a borehole by (a) determining the rock burden associated with at least a segment of the borehole; (b) determining a target Energy Factor EFT for the rock burden; (c) selecting an explosive material of known specific volume energy; and (d) calculating the diameter of the borehole needed to accommodate a volume of the explosive material sufficient to attain at least the target Energy Factor EFT.
The present invention further provides a computer-readable medium having computer-executable code therein for assigning positions for boreholes of predetermined diameter for use with a specified explosive material along a primary drill line along a bench of rock having a known density and having a rock face. Such a medium comprises (a) code responsive to user input defining a drill line having a start point and an end point; (b) code for determining a target rock burden BT for a hypo-thetical borehole having a height corresponding to the start point; (c) code responsive to data reflecting a model of the bench for defining along the drill line an incremental layer of rock having a mass less than BT and having an intermediate boundary and a distant boundary, determining the cumulative burden Bcum of the defined incremental layers, and the height of the layer at the distant boundary; (d) code for using the height at the distant boundary to calculate a revised BT; (e) code for causing the further execution of code (c) and code (d) until Bcum accounts for one-half BT; (f) code for setting and indicating the location of a borehole on the drill line between the intermediate boundary and the distant boundary of the last incremental layer when Bcum accounts for BT; (g) code responsive to said data for defining along the drill line further incremental layers of rock until Bcum accounts for BT; and (h) code for setting and indicating the position of the distant boundary of the rock burden associated with the borehole, and until the position of any previously accumulated layer exceeds the end point, for using the distant boundary of the rock burden as a start point and repeating the code of parts (b)-(g).
Further still, this invention relates to a computer-readable medium comprising (a) code responsive to user input defining a drill line having a start point and an end point; (b) code for determining a target rock burden BT based on a hypothetical borehole having the predetermined diameter at the start point; (c) code responsive to data reflecting a model of the bench, for defining along the drill line a progression of successive intermediate layers of rock each having a mass less than the target rock burden BT and being bounded by an intermediate boundary plane and a distant boundary plane and for each proximal layer (i) calculating a revised BT based on a hypothetical borehole on the last defined distant boundary plane and (ii) determining the cumulative burden Bcum of the defined intermediate until Bcum accounts for about one-half of BT and then setting and indicating the location of a borehole on the drill line in the last defined layer (referred to as the xe2x80x9ccentral layerxe2x80x9d); (d) code for defining along the drill line a progression of successive distant layers of rock, and accumulating the rock burdens of the distant layers until the total rock burden accumulated in steps (c) and (d) accounts for about BT; (e) code for setting and indicating the distant boundary of the rock burden for the borehole; and (f) code for using the distant boundary as a start point and repeating steps (b)-(e) until an incremental layer coincides with the end point.
The medium may optionally further comprise code for determining the position of the borehole between said intermediate and distant boundaries by interpolation.
In an alternative embodiment, the medium may comprise (a) code responsive to user input defining a drill line having a start point and an end point; (b) code for determining a target rock burden (BT) for a hypothetical borehole having a height corresponding to the start point; (c) code responsive to data reflecting a model of the bench for defining along the drill line an incremental layer of rock having a mass less than BT and having a distant boundary, determining the cumulative burden (Bcum) of the defined incremental layers, and calculating an average height of the incremental layers with each successive layer; (d) code for using each average height to calculate a revised BT for the hypothetical borehole; (e) code for comparing Bcum to BT for each successive layer and then for causing the further execution of code (c) and code (d) if Bcum is less than BT; (f) code for setting and indicating the location of a borehole on the drill line between the intermediate boundary and the distant boundary of the last incremental layer when Bcum is not less than BT, and using the distant boundary of the last incremental layer as a start point in response to code (e) when Bcum is not less than BT; and (g) code for comparing each incremental layer to the end point and for causing the execution of code (b)-(e) until an incremental layer coincides with the end point.
Optionally, the code (c) for calculating the average height of the incremental layers may comprise code for defining spaced parallel planes that define layer boundaries and taking the average height of the planes. Also, the code (c) for determining Bcum may include code for calculating the volume of the layer as one-half of the sum of the surface areas of the planes bounding the layer multiplied by the spacing between the planes, and for calculating the rock mass by multiplying the rock density by the volume. The code (d) may include code for determining the amount of the specified explosive material that would be loaded in the hypothetical borehole, determining the quantity of a reference explosive material corresponding to the calculated volume of a specified explosive material and calculating a target burden associated with the corresponding quantity of the reference explosive material. The medium may comprise code for scaling a mass of a specified explosive material by the relative magnitudes of the specific energies of the specified explosive material and the reference explosive material. Code for calculating the target burden may comprise code for determining a Mass Factor for the reference explosive material and multiplying the corresponding quantity by the Mass Factor.
The medium may comprise code for determining BT by determining an Energy Factor for the rock burden and relating the rock burden to the amount of explosive material that would be in the hypothetical borehole using the Energy Factor.
There may also be code in the medium for designating blast design constraints comprising minimum and maximum values for hole-to-rock face burden, hole spacing and at least one of a Material Factor and Energy Factor, and for determining the drill pattern characteristics of each borehole and comparing the characteristics to the constraints and indicating whether the constraints are met. Optionally, there may be code for determining the drill pattern characteristics of each borehole on a section-by-section basis for each borehole. There may also be code for analyzing deviations of drill pattern characteristics from the constraints to evaluate at least one of the drill line distance and drill line orientation relative to the rock face and reporting the evaluation.
Optionally, the medium may comprise code for identifying and reporting each borehole having an excess toe burden, and/or a swell or hollow in the rock face.
The invention also provides a computer-readable medium having computer-readable code therein for choosing at least one explosive material for use in at least a segment of a borehole having a rock burden associated therewith, comprising code for determining a target specific volume energy required for an explosive material relative to the associated rock burden; and code for identifying at least one explosive material that provides at least the target specific volume energy. Such medium may comprise code for referring to stored data indicating the specific energies of candidate explosive materials and comparing the data to the target specific volume energy. The data may indicate the specific volume energies of a plurality of blends of two or more materials. Alternatively, the medium may comprise data indicating the densities and specific mass energies of the various candidate explosive materials, and may comprise code for calculating the specific volume energy of a candidate explosive material and comparing the candidate specific volume energy to the target specific volume energy. Optionally, the medium may comprise code for partitioning the borehole into segments and determining rock burden and target specific volume energy for various segments of the borehole and separately identifying an explosive material for each segment. There may be code for determining the rock burden for the borehole and using a predetermined Energy Factor and the size of the borehole to determine the required specific volume energy.
Still another aspect of this invention relates to a computer-readable medium having computer-readable code therein for evaluating the suitability of a candidate explosive material of known specific energy for use in at least a segment of a borehole having a predetermined diameter and having a rock burden associated therewith, the code comprising code responsive to data indicating the specific energy of reference explosive material and a target rock burden to determine a reference Material Factor (MFR) for the rock burden associated with the borehole; code responsive to data for a candidate explosive material for calculating an adjusted Material Factor (MFA) corresponding to the use of the candidate explosive material in the borehole; and code for comparing the adjusted Material Factor (MFA) to the reference Material Factor (MFR) and for indicating the result. The medium may comprise code for calculating the adjusted Material Factor (MFA) by multiplying the reference Material Factor (MFR) by (MRef)(ERef)/(MEXP)(EEXP); wherein Mref=the mass of reference explosive in the section of the borehole; MEXP=the mass of candidate explosive in the section of the borehole; ERef=the specific mass energy of the reference explosive; and EEXP=the specific mass energy of the candidate explosive material.
Further still, the present invention provides a computer-readable medium having a computer-executable code therein for selecting the diameter of a borehole, the code comprising (a) code for accepting rock burden data associated with at least a segment of the borehole; (b) code for accepting a target Energy Factor for the rock burden EFT; (c) code for accessing data pertaining to the specific energy of an explosive material; and (d) code for calculating the diameter of the borehole needed to accommodate a volume of the explosive material sufficient to attain the target Energy Factor EFT.
The present invention further relates to a method for using a computer to assign positions for boreholes at a bench blasting site having a rock face, comprising inputting data indicating bench characteristics indicating at least bench height, bank angle, rock face configuration and rock density; inputting blast design constraints pertaining to spacing, hole-to-rock face burden, explosive material properties, desired borehole angle, at least one of Material Factor and Energy Factor; inputting a proposed drill line, start point and end point; and receiving a report containing proposed drill pattern characteristics.
In one embodiment of the invention, the report may identify boreholes having excess toe burdens and the method may further comprise inputting data indicating the placement of boreholes in the toe and receiving a report indicating positions for boreholes on the drill line. Optionally, the report may identify boreholes having rock face swells and the method may comprise inputting data indicating the elimination of at least one borehole position and the addition of at least one borehole on a swell between the drill line and the rock face.
Another method aspect of this invention relates to a method for assigning a cost-directed spacing to a borehole of predetermined diameter on a drill line along a bench of rock having a known density and a rock face, each location and borehole being subject to blast design criteria including a minimum spacing criterion, a maximum spacing criterion, and a minimum energy factor. The method comprises (i) proposing a compliant spacing for a borehole on the drill line with reference to at least one associated burden boundary; (ii) associating with the borehole a rock burden determined in part relative to the at least one burden boundary; (iii) determining a cost-based loading configuration for the borehole according to the method described above and recording the resulting compliant configuration (if any) and determining its associated cost; (iv) proposing a different compliant spacing with a corresponding borehole-boundary distance; (v) repeating step (ii), (iii) and (iv) for each different compliant spacing; and (vi) identifying the compliant spacing with the lowest cost (on a dollar per ton basis) compliant loading configuration (referred to as the cost-directed spacing). Step (vi) may optionally comprise identifying the location of the borehole and of the first and second boundaries of the rock burden associated with the cost-directed spacing. The method may optionally be repeated by using a boundary associated with the cost-directed spacing as a fixed boundary for the rock burden of a subsequent borehole on the drill line to assign a cost-directed spacing to the subsequent borehole. In one embodiment, the method may comprise first proposing in step (i) a spacing that corresponds to the minimum spacing criterion and proposing in step (iv) an incrementally larger spacing than was used in the previous steps (ii) and (iii). Alternatively, the method may comprise proposing in step (ii) a spacing that corresponds to the maximum spacing criterion and proposing in step (iv) an incrementally smaller spacing than was used in the previous steps (ii) and (iii).
This invention also provides a computer-readable medium having computer-executable code therein, comprising (a) code for selecting a segment of the explosive column portion of a borehole to be filled with explosive material; (b) code for determining the rock burden associated with the selected segment and for accessing data pertaining to blast design criteria comprising minimum and maximum energy factors and for accessing data pertaining to candidate explosive materials; (c) code for evaluating candidate explosive materials for use in the selected segment in order of priority until one is found that meets the blast design criteria (referred to herein as a compliant material), and for assigning the first compliant material to the selected segment, for assigning stemming to the segment when all candidate explosive materials fail to meet the minimum energy factor criterion, and for indicating whether all the candidate explosive materials exceed the maximum energy factor criterion; and (d) code that causes code (a), (b) and (c) to repeat for each segment of the explosive column. Optionally, there may be code for evaluating explosive materials in order of cost to generate a cost-directed loading configuration. There may be code for evaluating explosive materials in order of specific energy.
The invention further provides a computer-readable medium having computer-executable code therein for assigning a cost-directed spacing to a borehole of predetermined diameter on a drill line along a bench of rock having a known density and a rock face, the spacing and borehole being subject to blast design criteria including minimum spacing, maximum spacing, and minimum and maximum energy factors, the medium comprising (i) code for accessing data relating to the configuration of a bench and to blast design criteria comprising minimum and maximum energy factors and data comprising energy and cost characteristics of candidate explosive materials; (ii) code responsive to user input defining on the bench a drill line having a drill line start point and a drill line end point; (iii) code for proposing a compliant spacing for a borehole on the drill line with reference to at least one burden boundary; (iv) code for associating with the borehole a rock burden determined in part relative to the at least one burden boundary; (v) code for determining a cost-based loading configuration for the borehole as described herein and for indicating the resulting compliant configuration (if any) and determining its associated cost; (vi) code for proposing a different compliant spacing with a corresponding borehole-boundary distance and for causing code (iv) and (v) to repeat for each different compliant spacing; and (vii) code for indicating the compliant spacing with the lowest cost (on a dollar per ton basis) compliant loading configuration (referred to as the cost-directed spacing). The medium may optionally comprise code for indicating the location of the borehole and of the first and second boundaries of the rock burden associated with the cost-directed spacing. Also optionally, the medium may comprise code for using a boundary associated with the cost-directed spacing as a boundary for the rock burden of a subsequent borehole on the drill line and executing the code of parts (iii)-(vii) to assign a cost-directed spacing to the subsequent borehole.
The invention also provides apparatuses for assigning positions for boreholes of predetermined diameter for use with a specified explosive material along a primary drill line along a bench of rock having a known density and a rock face. The invention also provides apparatuses for choosing at least one explosive material for use in at least one segment of a borehole having a rock burden associated therewith, and apparatuses for selecting the diameter of a borehole. Each such apparatus comprises a computer processor; storage media accessible to the processor, for storing data and executable code; input means for delivering data to the at least one storage medium as described above; and output means for conveying data representing locations for boreholes, identifying an explosive material for use in a borehole segment and/or representing a selected diameter for a borehole, as appropriate.
The method of this invention may provide for reducing the cost of a blast pattern by (a) broadening at least one constraint by an acceptable degree to yield revised constraints; (b) proposing a hole position at a minimum spacing constraint; (c) evaluating a hole at that position for compliance with the revised constraints and calculating and recording the cost per unit burden mass for blasting at that position; (d) proposing another position for the hole at a different compliant spacing; (e) repeating steps (c) and (d) until the constraints are no longer met, then (f) evaluating the calculated costs per unit burden mass and indicating the spacing having the lowest cost per unit burden mass. The method may optionally include (g) proposing the next borehole at the minimum spacing from the previous hole; and (h) repeating the evaluation of steps (c)-(g) until the end point is reached.
Inputting data for the practice of any invention disclosed herein may comprise retrieving data from a memory medium, transferring data from an electronic surveying device, and/or entering data via a user input device. Setting data may comprise determining its value, e.g., setting a position or location for a borehole or boundary comprises determining, e.g., by approximate calculation, the proper position or location. Indicating data, such as the output of a computer program representing a location or position or the result of a comparison between two or more given values, may comprise displaying the data, recording the data for future retrieval and/or sending the data to another electronic device.